The transverse Mercator was invented by Lambert in 1772.
In this projection the cylinder touches a meridian along
which there is no distortion. The distortion increases away
from the central meridian and goes to infinity at 90 from
center. The central meridian, each meridian 90 away
from the center, and equator are straight lines; other parallels
and meridians are complex curves. The projection is defined
by specifying:

The central meridian

The latitude of origin

Scale along the equator in inch/degree or 1:xxxxx (-Jt), or map width (-JT)

The optional latitude of origin defaults to Equator if not specified.
Although defaulting to 1, you can change the map scale factor via
the MAP_SCALE_FACTOR parameter.
Our example shows a transverse Mercator map of south-east
Europe and the Middle East with 35E as the central
meridian:

we made the map illustrated in Figure 6.15. Note that
when a world map is given (indicated by -R0/360/s/n), the arguments
are interpreted to mean oblique degrees, i.e., the 360 range is
understood to mean the extent of the plot along the central meridian,
while the ``south'' and ``north'' values represent how far from the
central longitude we want the plot to extend. These values correspond
to latitudes in the regular Mercator projection and must therefore be
less than 90 degrees.